Low Expectations vs. Realistic Expectations

Of the many education quotes and adages I have floating around my head, one that consistently surfaces is George W. Bush's description of "the soft bigotry of low expectations." What does it mean when I have low expectations of a student? I see it sending two messages: (1) You're not smart enough to meet the high expectations that you see me holding other students to. (2) I've given up on you. The "I" in those statements could refer to me specifically, or to the "I" of a school or institution. I am honestly ashamed of some of the things I have let pass as "mastery" of content. It feels bigoted because it feels like I've already made the judgement that I've deemed them so incompetent that I don't think they can perform at a "real" level. It feels condescending because it seems like I'm lying to them to make them feel good about themselves. I feel like I'm no better than someone who writes them off in the first place. Maybe I even feel worse because I tried to earn their trust and then told them I didn't think they were good enough/smart enough/capable enough; the dagger hits closer to the heart.

But in trying to avoid the soft bigotry of low expectations, I keep coming back to two questions: (1) What are high expectations? What does it mean for something to be rigorous? How do I know I'm challenging a kid? (2) What does it actually look like to hold a kid to high expectations? I can give lip service to my expectations all I want, but what does it look like in practice to hold those high expectations?

And those are the easy questions. Like every good adjective, "high" is subjective. I think most people would agree that high expectations for sixth graders are (and should be) completely different from high expectations for 12th graders. Sixth graders are in a different place emotionally, developmentally, and academically. You wouldn't stick a sixth grader in a calculus class and pride yourself on how well you're challenging the kid. I would make the argument that my students are in a different place, at least emotionally and academically, than other 10th and 11th graders in, say, Palo Alto. So what does it mean for me to hold high expectations for my students, given that those expectations will be different from high expectations at Palo Alto high schools? If my expectations are different for kids at my current school than they might be at another school, am I inherently succumbing to the soft bigotry of low expectations, especially because my students are almost all children of color from low-income families?

Then there's the piece about what happens when you try to hold students to high expectations. Let's say I try to be that tough teacher and hold my kids to agreed-upon standards of high expectations. For example, what if I started teaching my geometry according to IB standards? Kids would fail my class. Maybe that's being pessimistic, but I also feel like it's realistic. I know it's my job as a teacher not only to set high expectations, but to help kids meet them. At some point, though, the kid has to do some work. I don't know enough about scaffolding or adolescent development or motivation or any of that to get students to put in the level of work that's required for them to compete at the same level as those Palo Alto kids. And even if I did know, I don't think I have the energy. If I compare it to sports, I could set a goal for myself of running a marathon in 2 weeks, which is for sure a high expectation. Maybe--maybe--I could even meet that goal if I was motivated enough to spend all of my time training and drop everything else I'm doing with my life. But that's unrealistic. The response to that is, "Set a goal to run a marathon in 6 months--that's reasonable." Great, but I only have a limited amount of time with my students. The marathon they have to compete in is coming in June (or May, if we're talking about state testing, or February if we're talking about the High School Exit Exam), so even if they're completely out of shape, they still have a limited amount of time.

Happy new school year. How do I set expectations for myself that aren't setting me up for failure and/or burnout?

Another Addition to the Banned List

Awhile back I wrote a post detailing some of the phrases and teaching strategies I would like to see banned from our nation's math classrooms. There are always little things that make me wonder why a teacher chose a certain strategy ("When you multiply by 1 it stays the same because 1 looks like a mirror, so it reflects back the number." Do you think that kids couldn't reason out why multiplying by 1 doesn't change the value?). But there's been a big one coming up lately that's kind of been driving me out of my mind.

"Reducing" Fractions
I understand why we say reduce, especially when accompanied by the phrase "to its simplest form," but I what do you think of when you think of something being reduced? You think of something lessening, you think of it having not as high of a value, you think of it not amounting to as much as it originally did, etc. Kids in the US have a lot of trouble with fractions, particularly with the concept that fractions represent a specific type of comparison of a part to a whole. When one 'reduces' a fraction, an important conceptual aspect is that even though the numerator and denominator change in a way that lessens their respective values, the fraction as a whole does not change in value at all. To say that a "reduced" fraction is equivalent to its original form is kind of an oxymoron--I'd love to see a sale where the reduced prices are equivalent to the original prices. When we use the word "reduced," are we surprised that kids don't understand fraction equivalence?

So what's the better option? Simplify. I love this word for a few reasons. First, it's preferable to "reduced" because it makes sense that a simplified version of something still has the same meaning/value as the original--now it's just in an easier-to-understand form. Second, I want kids/people to realize that any value can be represented in infinite ways, but we prefer some ways because they are easier to work with, compare, interpret, manipulate, or use in a given situation. Would you want to go to a restaurant where a sandwich cost $1-3^(0!)+300/(4x10)? Mathematically, there's nothing wrong with that price; realistically and emotionally, it's just annoying. If we simplify the price (not reduce because the restaurant still wants the same amount of money), we get a value that's more useful for our brains because the simplified value is easier to compare to what we already know about sandwich prices. Is  $1-3^(0!)+300/(4x10) expensive? Cheap? I have no idea until I simplify it to a value that looks like the other values I could compare it to. It's the same reason why we change the form (not the value) of a fraction when we need to add it to another fraction, or why we sometimes factor a quadratic into a binomial and sometimes multiply a binomal into a quadratic, or why we convert linear equations into slope-intercept form. Math is only useful if we can make meaning from it, so we manipulate values, expressions, and data sets until we can mold them into something that makes the meaning easier to find. We're not reducing anything when we rewrite 32/40 as 4/5; we're simplifying it into an equivalent form that's more useful.

Happy Holiday

About a month ago, I got this in the mail:

Unfortunately, I did not heed the urgency of Atlas Pen & Pencil Corp's envelope and once again Pi Day has snuck up on me before I even realized it. But thanks to the nerdy blogs I read and even nerdier company I keep, next yet I will be prepared with an arsenal of Pi Day-related fun. Here, for your celebratory enjoyment:

This is so freaking cool. Is it worth the time to cut out all those apple and pie crust slices? Probably.

Thanks to this useful allocation of government time and money, I now know what "squaring the circle" means. Dear State of Indiana: do not try to legislate mathematics because it is above the law. Or, more accurately, mathematics is the law.

My students were super-excited that they are going to graduate in very important year, Pi Day-speaking. For the only time in any of their lives, it is their senior year of high school when we will get to have a completely legitimate Pi Day on 3.14.15. Some were more excited than maybe they should have been.

Finally, in case you forgot to celebrate today, fear not. There are many more mathematical holidays yet to be turned into Hallmark occasions and/or crappy math lessons. Some seem to be taken a little more seriously than others.

Happy 'Sorry Your Math Teacher Didn't Serve You Baked Goods (Mostly Because She Wants You To Learn The Correct Spelling Of The Greek Letter)' Day!

Sometimes high schoolers are cute

I already gave my final exam, so there is now some down time for my kids who don't also take algebra. I had nothing planned for today, so I broke out the pattern blocks. This group decided they wanted to work together to make one giant, cooperative pattern.

Now they are sitting in almost complete silence, completely engrossed. Every once in awhile they discuss what color or shape to place where. Who says that kids today are only interested in constant, passive stimuli?

I love these kids.

What kind of filling do you think is in the middle?

The Venn Piagram!

A geeky math joke, I know. But it makes me excited for Thanksgiving.

What My Gang Affiliation Would Be

After school today, a couple of students were trying to convince me to let them give me a henna tattoo.

M: "We could write 'I love math' on your arm."
Me: "I'll think about it."
E: "We could write 'M.O.B.'"
Me: "No, you could not write that."
E: "Math over biology!"

Is She STILL Talking about Asilomar?

Yes, I am. Partly because the conference is always amazing and inspiring and partly because I just now finally got a chance to upload these pictures. This is what the Asilomar conference grounds look like. So even if the conference had been professionally useless, at least it legitimately counted as a weekend away.

By the way, these are actual shots of the conference grounds and Asilomar State Beach, which borders the conference grounds. Unfortunately there was so much rain this year that I didn't get my usual session-long walk on the beach. I opted for napping to the sound of crashing waves instead.

Asilomar Reflections: Jo Boaler

I should preface this by saying that I have a pretty major bias in favor of Jo Boaler's work. This is not for reasons of blind faith, but because of how I have been "raised" as a math teacher. The teaching methods I've learned and successfully used have been the study of her research, not to mention that most of my math teaching heroes were either her research subjects or worked/studied with her at Stanford.

But regardless of all my biases, Dr. Boaler always has interesting things to say. Her doctoral thesis was a study of two schools in England, one using traditional mathematical teaching methods and one using "reform" methods. The differences between the two schools were startling. The students at Phoenix Park (the reform school) had much better conceptual understanding as well as a more positive and productive attitude towards math than the students at Amber Hill (the traditional school). There's more in her thesis if you want details. The same study was repeated at similar schools in the United States. The results were published in the Phi Beta Kappan under the title "When Leaning No Longer Matters," if you're interested. And you should be interested.

A couple years ago Boaler was giving a talk about her England study and a man in the audience asked if she knew what had happened to those students. It turned out that he happened to be a rich oil tycoon who was willing to fund her to go back and find out. So she did, and presented the results at Asilomar. Again, the results were fascinating. Part of her general thesis is that math classes literally traumatize and leave life-long scars on students. All math teachers know this anecdotally from the reactions we get upon telling people our profession. "Oh, I hated my high school math classes." "Wow, you're brave! That was my worst subject." "I actually like math" (as if that should come as a surprise).

So how would adults feel about math if they came from a reform background? The Phoenix Park students (now in their mid- to late-20's) definitely had a more positive attitude toward math, both as a general interest and in how they used it in their jobs. Dr. Boaler asked both groups what they do when they come across a math problem they can't solve. Most Amber Hill (traditional) students said something to the effect of, "I would ask someone who's good at math." One Phoenix Park student responded, "Why wouldn't I be able to solve it? If I didn't get it I would just keep working until I did." As a math teacher, that warms my heart. Even my "smartest" kids struggle with persistence more than pretty much everything else. I definitely believe (and lots of research has shown) that persistence and belief that you are capable of succeeding in a math problem are key factors in how well people do on tests.

For those who don't care about the wishy-washy how do people feel about math, Dr. Boaler had some pretty stark numbers. Both groups of students she interviewed were coming from similar socioeconomic backgrounds. For the Amber Hill students, their range as adults got wider, meaning that some did move up to higher levels, but just as many fell. The range of the Phoenix Park students stayed the same width, but moved up, meaning that very few, if any, students ended up at a lower socioeconomic class than their parents. That in itself is impressive and also has serious implications for raising low-income communities out of poverty.

In the age of high-stakes testing (I know this is how so many sentences in education writing start off...), there is so much pressure to teach to the test, to jam facts into kids' heads so they'll remember them for 2 hours in April. It's so tempting to teach them stupid tricks and drill them to memorize formulas without any reasoning behind it. It's definitely easier to write a worksheet of 100 drill-and-kill problems than it is to come up with one good groupworthy one. But Dr. Boaler's talk was definitely the reminder I needed that teaching conceptually, getting kids to talk about math, and helping them value different ways of seeing are such worthwhile goals. It's hard when my standardized test results don't seem to demonstrate how smart I think my students really are, but if they can go into their adult lives believing that they are capable of math and having the habits of mind to think mathematically, I'm a little less worried about their CST scores.

Asilomar Reflections: Lucy West

Don't make fun of me for being nerdy, but the California Math Council (North) annual conference is one of my favorite events of the year. Who can pass up a weekend at Asilomar State Beach talking about interesting math and teaching topics with some of the biggest names (and nicest people) in math education? I can't believe I missed it last year.

This year, as is true for the other times I've gone, I learned a LOT. The Friday night keynote speaker, Lucy West, was particularly impressive. She talked about academic discourse and its necessity in a math classroom. This is already one of my core beliefs about teaching math, but I liked her talk because she confirmed/affirmed some of the things I already do and also raised a couple of new points that I hadn't thought of.

Things I'm already doing:

• First of all, her thesis was that if kids talk about math, they learn more and better. As I said, this is one of my core beliefs and I structure pretty much everything in my classroom around this. How can I get them to talk? How can I get them to listen? More importantly, what can I give them that's actually worth talking about and listening to? Ms. West emphasized the role of both teaching and listening, both of which I try to scaffold on a daily basis. I think my kids are getting better at it (although their listening could use more work than their talking). They're taking a group test even as I write this and I'm generally happy with the quality of their conversations, especially given that they're 9th graders who have only been working on this for 3 months. 

• There were certain "teacher moves" that Ms. West pointed out in videos and I was really happy that I already do most of them. I'm a huge fan of "Talk with your partner/team, now share out with the class what you talked about" and the "Who can rephrase what _____ just said? ... Great, now what's another way of rephrasing it?" I'm also into the accountability move of "You're not sure? Who else wants to explain it, and you'll rephrase what they said."

New ideas (sort of): 

• We all know that teaching is a painfully isolating profession. Even at my school which supposedly is highly collaborative, I am the only teacher teaching my subject, so I have nobody to plan or debrief with. I have maybe seen 2 minutes all year of another teacher teaching. Maybe. Ms. West compared teaching to surgery: if you watch Grey's Anatomy, all the doctors (interns, experts, everyone) are always excited to be in on each other's surgeries. When there's a new procedure, they all want to help out with it or at least be in the viewing gallery. I learn SO much when I watch other teachers or when other teachers watch me. I would love to see another teacher try out some hot new activity or teaching move, even if it fails. I know that if I hadn't been in a teacher ed program where we looked at what it means to have math discussions (or if I never had a teacher ed program in the first place), I would still be lecturing everyday because that's the only kind of math teaching I ever saw. I didn't know there were other options out there--and I imagine that many teachers don't. Who knows what else I'm missing by not talking to or seeing other teachers. All those teacher moves that I learned are things that I saw other people doing. And it's so crucial that observations happen frequently. I always meet math teachers at conferences who are still lecturing all the time and who can't even conceptualize how to incorporate discussion because they're only hearing about it for 45 minutes at a conference. They go home and try to use the ideas, but it fails because not only did they get such a small glimpse, they have no one to support their trials (and inevitable failures). To see what it actually looks and feels like--and how to do it well--you need to see something over and over and in different contexts. 

• This brings up another of West's points. A lot of the terrible ideas in math education came from a really good place, but ended up being adapted into something awful. Her example was "the walkthrough," a practice that had positive origins. Teachers and administrators decided together what kinds of things might be observed for in a snapshot observation, and they followed observations with conversation. This is quite the opposite of walkthroughs I have experienced where administrators would come in for >5 minutes with a checklist: Are there objectives on the board? Are kids using the textbook? Are there state standards on the board? Then they would leave. These things (and many of the other things they look for) are related to components of good teaching, but on their own do not create or indicate good teaching. The evolution of the walkthrough is like a bad game of telephone. I see this happening all the time with groupwork and complex instruction (structures and philosophies that are very near and dear to my heart). Someone watches a video or sees a good math problem at a conference so they take it back to their school. And it flops. But there's so much that goes into good groupwork that takes so much time, so many deliberate decisions, so much more than just trying to imitate 10 minutes of video. I know that I never would have learned all that I have if it weren't for two specific supports. First, at my previous school my teaching schedule was designed so that I could observe a veteran teacher teaching algebra during my prep period. I got a day behind in the curriculum so I could see Estelle teach a lesson on Monday and then I would teach it on Tuesday. We would have already talked through the misconceptions, the pitfalls, the strategies, etc. so that I had an idea of what I was getting into. It's so different seeing a lesson in action than it is reading about it in a lesson plan. The second support is having other expert teachers coaching me in the moment while I'm teaching. At my old school department members came in for particularly challenging activities (challenging for the teacher, not necessarily the students) or if I needed help with a certain group of students. Sometimes other teachers would just bring their grading over and sit in my room to work. This happened for both veteran and new teachers. Currently I have an amazing coach who sometimes comes into my class (not as much as I'd like) and whispers little moves or ideas as I'm teaching. AND we plan curriculum and debrief together. In the first few weeks this year when she was in my room almost every day, I learned so much more than I did in all of last year. I can't come up with good ideas and teacher moves out of thin air. Learning about teaching has to come from seeing other teachers teach. 

• Sticking to the medicine analogy, doctors have a common language with which to discuss their practice while teachers... not so much. When I talk to other teachers, even from my own school or department, we have different ideas of what a "warm up" is or what it means for kids to work in groups. It's fine for different teachers to be doing different things, but it's a huge challenge to have a conversation when you're talking about two different things but using the same words. 

• This next idea was new to me, but made perfect sense. Ms. West said that when she goes into a new school, the quickest way to predict what classroom discourse will look like is to listen to how the teachers talk to one another. If teachers never talk together about math (or academic content), it's unlikely that students will. If teachers are competitive and protective of information, students will be too. This totally makes sense to me based on the two schools where I worked. Ms. West asked, "Think about the meetings you have to go to--do you look forward to them?" At my previous school, definitely yes. Our Thursday algebra meetings were the highlight of my week, partly because I really liked the other people I work with, but mostly because I learned a lot from them. We spent a lot of time doing math problems, looking at curriculum, and sharing ideas. Our kids did the same in class. The conversations at that school--especially in upper grades where kids had been working this way for years--were mind-blowing. The math was deep and the their communication was gentle and caring. At my current school, discussions look different. Most meetings are highly structured with limited space to discuss ideas. There's a time and place for everything and no time outside of that. We get a lot done in a relatively small amounts of time. The same is absolutely true of our classes. We have 2 months less of class time than other schools, but get higher test scores. Most teachers use highly-scaffolded lesson plans, worksheet formatting, etc. Everything happens in a very measured way, but we're always moving. I'm not saying that one approach is better than the other, (although I do have a personal preference for the former), but it is interesting how the students mirror the adults. 

• Ms. West stressed the necessity of adults doing math together, which our department does not. It's unfortunately first because hey, math is fun. More importantly, it's unbelievably powerful to consider the experiences our students have in class. I'm always amazed at the behaviors I revert back to whenever I have to do math in a group. For example, I went to a session at the conference that was just doing interesting geometry problems (remember, I'm a geometry teacher). The two people I was working with are friends I met at Stanford who I continue to be very close with both personally and professionally. In general I am comfortable around them and confident in my abilities, but when got stumped by a system of equations (which was an algebra II level of difficulty) I instantly turned to hide my paper and pretend I knew what I was doing. I absolutely know that my friends would never look down on me for asking a question or for having trouble with that problem, but the math anxiety came out so quickly. If that's what's happening for me, a "mathematically successful" adult doing problems for fun with my friends, what must be going on for some poor little freshman who's never felt good about math and is working with kids s/he barely knows? It's so rarely that we as adult math teachers come across problems we don't know how to handle, but our students do every day. If there's ever any questions that students' emotions impact their learning, the answer lies in working on math problems with other teachers. 

During Ms West's talk I sat with many of my former colleagues, most of whom are no longer still at our school. We shot each other a lot of meaningful glances as Ms. West more or less named key features of how we worked together and what we worked toward. I am constantly reminded of how lucky I've am to have been a part of this group, even if it was only for a year. The things I learned and the connections I made continue to inspire and inform my teaching. My current department is great, but I don't always feel like we're in the same place, have the same values, or are working toward the same goals. How do I get back to a place where I believe so deeply in what my department is trying to do? 

Gift Ideas

Just in case you're having trouble figuring out what to get for that special someone (i.e. me):

http://theorymine.co.uk

Banned

I very much understand that there are many different ways to teach math well. I see other teachers sometimes and think, "WOW... but I could never do that." It's often a matter of personal style and most importantly, assessing and meeting the needs of your specific kids. Somewhere in all of that, however, are best practices that we (should) learn in ed school or from each other. In the same vein, there are many different ways to teach math poorly. Somewhere in all of that are what you might call anti-best practices, certain ways of teaching that do more to harm student learning than to support it. Besides all the "teacher moves" that don't work, there are some content-specific anti-best practices I've seen over the years that make me want to pull my hair out every time a kid mentions one.

The following is a list of phrases, algorithms, and "tricks" that I wish would be banned from every math class.

Cross Multiplication
Cheers to the genius who decided that it would really help kids to draw a big X through an equals sign and multiply the numbers on the ends. Yes, cross multiplication "works" in the case of a straight proportion with one unknown, but that's a very specific case. Kids like cross multiplication because it's easy to remember (although the dividing at the end gets confusing), but what they don't remember is when you should actually use it. Try this experiment if you have a middle schooler or high schooler handy: if they remember cross multiplication, then give them three problems, one where two fractions are set equal to each other, one where the fractions are added, and one where they are being multiplied. You'll be amazed at how many kids draw an X on all three problems. This is so common that my lesson on cross multiplication (and why you're not allowed to use it in my class unless you can explain to me why it works) is based on someone raising their hand and making this mistake. And it always happens. Kids don't understand why it works, so they don't know how or when to use it.

One of the biggest things I want my students to learn from my class is that math and math steps should always make sense. I teach them "fraction busters" which is essentially a long version of cross multiplication. Fraction busters involves multiplying by all the denominators in an equation in order to "bust" the fractions and get something more reasonable to work with. I push my kids to say that we're multiplying because fractions are division and we want to undo the division by doing it's opposite. If there's a reason behind their math steps (besides "It's what my old math teacher taught me") then there's a very good chance that they're doing something right, and an even better chance that they'll understand how to fix it if it's wrong.

Fun story: last year another math teacher and I literally sprinted across the school trying to find the algebra teacher to make sure he didn't teach cross multiplication. I like to think it was like a movie when Maura found him, that she busted through the door (no pun intended) and dove in slow-motion to knock the whiteboard marker out of his hand before he could draw an X through the equals sign.

"Cross Cancel"
Honestly, I don't even know what this means, but it always comes out of some kid's mouth when we're talking about fractions. I think it has something to do with multiplying and simplifying fractions? In any case, if I don't know what it means and kids can't explain it to me, it's obviously not helping them learn. It's just another "rule" that has no sense-making behind it, so kids mess it up.

FOIL
Oh, binomials, the way you torture Algebra I students is amazing. FOIL (first, outer, inner, last) is definitely the way I learned how to multiply binomials and definitely the way I continued to do it until a couple of years ago. Again, this is another "rule" that kids don't understand the reasons behind. It's a little harder to mis-apply  than something like cross multiplication, but it's detrimental because it's extremely limiting. I was a good math student, and I definitely did not understand for a very long time how to multiply a binomial with a trinomial (let alone larger polynomials) because it didn't fit the FOIL pattern. Another significant issue is that because FOIL has no logic behind it, there's no logic to un-FOILing AKA factoring, which is consistently one of the most difficult topics in Algebra I.

The solution? The best way I've ever seen this taught--and what I continue to use--is an area model. Here are some examples:

They make a lot more sense to kids, and work well for any kind of multiplication of polynomials because you can make the generic rectangles any size. They also make factoring unbelievably easier. Learning about generic rectangles was life-changing for my own mathematical understanding and for the way I teach algebra.

"A negative and a negative is a positive"or "A negative and a positive is a negative."
What's the key word in these phrases? Oh, there isn't one because it's completely ambiguous. What does "and" mean? To make these "rules" correct, "and" should mean "multiplied or divided by" but of course kids don't remember that. You'd be amazed at how many kids think -5 + 10 is either -15 or -5. Multiplication of positive and negative numbers gets a little more complicated with regard to the sense-making aspect--can YOU explain why two -5 x -5 = +25? But the sense-making around adding positive and negative numbers is not difficult, so if we can at least start there hopefully there will be less confusion with multiplication/division.

a^2+b^2=c^2
No, I don't want to get rid of the Pythagorean Theorem. But who thought it was a good idea to name the sides of the triangle a, b, and c? How do those help, especially when it doesn't matter which of the legs is a and which is b? I got a semi-angry email from another math teacher this year asking why we'd teach it that way. The answer: I don't. I teach it by first looking at the geometric representation, so I want the kids to be able to verbalize something like, "The areas of the two smaller squares add to equal the area of the bigger square." Algebraically, I teach it as "Leg^2+Leg^2=Hypotenuse^2".

"Cancel out"
Another terribly ambiguous phrase. Kids love crossing things out (who doesn't? It feels so satisfying), so they love saying that things cancel. But "canceling" is another word that means nothing, at least when it comes to sense-making and giving reasons. We teach that adding and subtracting from both sides of an equation is "making zeros." Simplifying a fraction is "making ones." Squaring and square roots "undo each other." Because that's exactly what's happening. Math isn't magic, so numbers and symbols shouldn't just vanish with no explanation. If kids can make sense of these very fundamental identities, they're set up to understand all kinds of algebraic manipulation and solving, as well as new operations and their inverses. And kids still get to cross stuff out.

There are two main ideas behind this post:
1. For kids to effectively learn math, it has to make sense. Seemingly arbitrary rules do not help them make sense of anything and only serve to deepen their belief that math is some confusing, magical subject that's only accessible to a select few.
2. Unlearning a misconception is exponentially more difficult than learning something right in the first place. And what's tricky is that misconceptions are often applied in a way that gets a correct answer. So kids who I meet in 9th grade who have been cross multiplying for 2-3 years get frustrated because they've been able to solve all these proportions correctly, not realizing that the other ways they've been applying it are where the real problem is.

I'm not completely innocent regarding all the anti-best practice above. I did teach things that way or would have taught them if I'd never seen an alternative. It makes me wonder what else I'm doing wrong that's driving upper-level teachers crazy...

Speed Demon

More thoughts on how teachers spend their time:

I started grading projects at 11:30am this morning and finally finished the grading portion at 5pm (plus about another hour of entering grades, sorting papers, creating and emailing lists of kids who need to revise, and all that other business mess). I'd say that I took about a total of 45 minutes in that time for little breaks--checking email, texting, refilling my coffee, using the bathroom due to coffee refills. So that comes out to 330 minus 45 minutes of grading for 105 projects. Amazingly that divides into an average of 2.71 minutes per project. If I add the other 60 minutes of business, that means I spent an average of 3.28 minutes per student today.

In some ways I am proud of this; I expected it would probably take me about 5 minutes per project, which comes out to 8.75 hours. On the other hand, I did not finish early enough to take advantage of today's 90-degree weather so the day was already a wash in terms of my emotional health. On the other hand (yes, I have 3 hands), my kids spent about a week working on these projects (at least 5 hours of time per kid, I reckon). And in return I spent between 2.71 and 3.28 minutes thinking about each of them. There some sort of proportional reasoning lesson in there.

I don't know what to make of this data. What I do know is that I have about 2 more hours worth of work tonight before I'm ready for tomorrow.

The Next Alexander Calders

Back in January when our school was in intersession, one of the courses was a math class just for 9th graders. Students were selected to participate, with the goal that it would support struggling students. Of course, if you take the 20 kids who need the most help in math, they're probably also the kids who like math the least and are least interested in doing it for three hours straight every morning. Fortunately, the teachers are two of the most amazing math teachers I've ever met (and with no exaggeration, probably two of the most amazing math teachers in the country) who planned a class specifically around helping students grow their mathematical confidence and skills by providing them opportunities to engage with deep, difficult mathematics.

In addition to many other tasks and problems, the students had an ongoing art/math project that lasted for the entire month. Each student built a perfectly-balanced mobile using foam shapes. This was much more than just playing around with balance, but used some serious math that would challenge the average adult. First, students experimented to figure out the relationship between two objects of different weights and where they would have to be placed in order to maintain balance. Then they chose a theme for their personal mobiles and designed at least four pieces. The requirements stated that (1) all pieces would be built out of basic shapes (quadrilaterals, circles, triangles) and all shapes must be used at least once, (2) no more than one stick could have its string balanced in the middle, and (3) they would have to make all calculations and prove mathematically that their mobile would balance before they could actually start cutting out their shapes. Like I said, that's some serious math.

The results were amazing. The students were more creative than I could have imagined. They got really into the artistic aspect and nobody tried to "go easy" by just picking easy shapes. Some of the math they ended up using, and seriously improving at, included:
-Number sense
-Measurement
-Converting fractions to decimals
-Calculating areas
-Spatial reasoning
-Proportional reasoning (the balancing relationship is directly proportional)
-Working backwards
-Measuring area versus length
-Productive disposition
-Connecting mathematics to real-world applications

Okay, now the good part: the finished products. My mind was blown by these.

The Spider-Man mobile (the artist refused to pose with it, so I took her place):


Music theme: a radio, iPod, headphones, and a radio


Food theme: pizza, ice cream, pop, and a Hershey bar (obviously made my a growing teenage boy)


Three girls made this one as a gift for one of the teachers who is pregnant with twins


Another mobile for the twins' nursery


Clearly made by a teenage girl, but still extremely impressive in the way she used basic shapes to create very not-basic objects


Transportation theme: a rocket, a boat, and a tank (carrying Spongebob and Patrick, of course)


Lots of happy faces


This is one of my favorites (shhh, don't tell the kids). Partly because I love the Japanese monsters, partly because she used the shapes so creatively, partly because she made it for her brother, and partly because this is from a kid who I've never, ever seen excited about math (until this)


There are a couple others that are truly amazing, but I decided not to post them because my pictures include students. Email me if you want to see the girl whose theme was European landmarks.

Found: Note

Although most kids these days send their in-class communications via text message, there are still those hold-outs who pass notes in class. I guess it's less detrimental to get a piece of scratch paper taken away than it is your iPhone. And it's generally more entertaining for me, the teacher. I feel too guilty to go through text messages, but I definitely do not feel guilty reading whatever note they were passing.

So far I have two favorite lines from notes I've found.

1.
Girl 1: "I can't tell if he likes me."
Girl 2: "You should talk to him."
Girl 1: "Who are you, Dr. Phil?"

2.
Girl 1: "Guess what? Yesterday I got my geometry test and I got my first A+ and A!"

On one hand, Note #1 is way funnier (especially if you know the two girls who wrote it). On the other hand, if they're passing notes in math class, I'd much rather that they're talking about what they got on their latest tests.

Just Instructions

It's true, the worksheet I gave in class today was a little more text-heavy than usual. The first bit of text was almost four sentences long and contained important mathematical information. The general format of the worksheet was this:

-Approximately four sentences of text reminding them what we did yesterday and explaining how the picture below is a picture of what we did yesterday.
---Picture/graph---
-Problem #1, asking students to draw triangles on the above picture
-Problem #2, asking students to write down relationships they noticed between the triangles they just drew.

I am not sure exactly what was confusing about this layout (I spend a lot of time thinking about layouts and formatting that make the content more accessible), but something told the kids, "It's okay, you don't actually need to read the questions." I think that in math classes in general, kids are used to seeing problems where they just know what they're supposed to do. They see an addition problem and they add it. They see a multiplication problem and they multiply it. They see an equation and know they have to solve for the variable. So it's a paradigm shift when they see a page where numbers and processes aren't clearly laid out for them. Because my students work in groups 99% of the time, the synergy of four students usually is enough for one person to ask, "Wait, what are we supposed to do?" Then someone realizes that they could figure this out by reading the instructions. It also helps (or doesn't help, depending on who you ask) that I answer at least half of their questions with, "Well, what did you read in the instructions? What does it say you need to do? Read it back to me. What does that mean you're looking for?" I even prompt every group conversation to start by someone asking, "Who wants to read?"

Getting back to today's worksheet, in 6th period (Lettuce) I came across one group that had skipped drawing the triangles and trying to answer the question about the triangles. Although I cannot say for sure, I'm pretty sure they skipped the reading part too. I checked in with them, asking how they could notice relationships between the triangles when they hadn't actually drawn any triangles on their paper. All three students in the group stared at me blankly.

Student: "What are we supposed to be drawing?"
Me: "What did you read? What does it say you need to draw?"
Student: "Where does it say we need to draw something."
Me: "In question 2. What did question 2 say?"
Student: "Oh, we skipped that. We thought it was just instructions."

Interesting. The fact that the text in question 2 was "just instructions" was exactly the fact I was trying to point out to them.

The Pythagorean Leash

One of the most important things I think that I can train kids to do in geometry class is to draw pictures. Any time they come across a word problem, they need to be able to translate it into a picture because 90% of the time, word problems aren't that challenging if you can get the picture right. Furthermore, translating word situation into a mathematical model/equation is a skill that will be legitimately useful for them long after they've left school (just ask the history teacher who came to me the other day trying to figure out how many points his final exam should be worth).

I'm also a huge fan or word problems because I love writing ones that will elicit awesome drawings. I get a sad when I have to write ones like "Point A is the midpoint of BC and angle JKL is bisected by ray KM and WU is the perpendicular bisector of XY such that WU is the altitude of triangle WXY..." because those pictures just end up being shapes with a lot of letters on them. The good problems are ones like I put on a recent quiz. The gist is that Julian/Karla (obviously I always use student names and weirdly, the kids--even in high school--still LOVE it) is walking his/her pet spider/pig and I give dimensions about the height that Julian/Karla is holding the leash and how far the pet is in front of its owner. It's just a simple pythagorean theorem problem, but the pictures are the good part.

The pig in this first one looks so happy and carefree. Maybe because he is out for a lovely walk instead of taking a geometry quiz.

 

My favorite part of this one is that the pig is explaining all of the answer. I wonder whether the speech bubble was planned in advance or a last-minute add-on. Either way, brilliant.

 

Here, I can't figure out if this student named the pig because (1) she had too much time left and got bored, (2) she was killing time trying to figure out the problem, (3) she really needed to have a name for the pig in order to make the problem feel accessible, or (4) she knows more details about Karla's pets than the rest of us do (this student is one of Karla's good friends).

 

Not to judge students' artistic skills, but this is the least pig-like pig I saw on anyone's paper.

 

I think I have really gotten through to this student when I've stressed the importance of labeling everything you know about a picture. I'm so glad he labeled everything because I definitely would not have known which was the spider and which was Julian.

 

This is my favorite picture out of any of them. In case you can't tell (sorry, I took these with my camera phone as I was grading them), the spider-walker's t-shirt says, "Hi! I'm Julian."

 

And I'll conclude with a kid whose math skills are doing fine (he got the problem right, at least), but who could use a little work on his reading comprehension.

 

Math in the U. S. of A.

In an attempt to get kids focused on problem-solving strategies without getting caught up in the details of formulas and procedures, over the last two days I've given kids tricky area and perimeter problem with no numbers. Here's an example:



S., a student whose dry humor never fails to make me smile, was getting frustrated with one of the problems. In exasperation, he literally threw his hands up and asked, "Why can't we just use numbers--like Americans?"

Oh S., at age 14 you've already identified the key difference between American and other mathematics education systems. How did you figure it out so quickly when it's taken researchers years?

The Awkward Triangle

In our recent area unit, we've talked a lot about how "every shape is a rectangle" and if they can just find the rectangle, they can find the area. For example, a triangle is half of a rectangle, so because we can find area of a rectangle by base x height, we can find the triangle by dividing the rectangle's area in half. On the test there was a question that showed an oblique triangle, noting that the triangle does not look like 1/2 a rectangle, so why does bh/2 work?



Here is the best description I've ever gotten as to why this kind of triangle is tricky:
"The triangle is more pointy so it's confusing. This works because no matter what kind of triangle it is it will still have another rectangle that fits its awkwardness."

I do love awkward triangles. Almost as much as I love the awkward things kids write on tests.

Appreciation Friday

Dear Mrs. L,
You are officially the best math teacher ive ever had! Everything you have taught was explained thurroly (don't know how to spell), which I am so thankful for.
Becaus I appreciate you so much I've put a sticker of a ninja on my letter! Keep up the good work.
Sincearly, N.L.

----------------------------

Dear Ms. L,
I am writing this letter because I appreciated when you helped me when I needed help. When I didn't get the formula (b1+b2)h/2=A you made it clear & now I got it. I didn't know where the 2 was coming from. So thank you Ms. L, I appreciate you lots lots lots.
<3, N.S.

PS. I also appreciate how you don't make us memorize formulas. We make our own formulas (that work).

Best/Worst Assessment Ever

My roommate came up with a pretty ingenious costume for me this year. See if you can guess what it is (sorry, I didn't take any pictures):
I wore an outfit that was all pastel (green plain bermuda shorts and a pink sweater vest), my hair in pigtails with ribbons, and freckles on my face. On the front of my sweater was a big sideways "V". On the back I had a sign that said "<90degrees".

Get it? I was a cute angle! Hahahaha. Okay, not that funny, but appropriate for a geometry teacher. Nowhere near as good at the statistics teacher who came as a Mean Girl (entry in her Burn Book: "Didn't find y-hat"). As expected, some kids thought my costume was funny and most rolled their eyes. However, there was one reaction--that many had--that I didn't expect.

Kid: "Ms. L, what are you?"
Me: "Guess"
Kid: "Ummmm... a little kid? A nerd?"
Me: "What's on my shirt?"
Kid: "Tape?"
Me: "Yeah, what shape is it?"
Kid: "A 7?"
Me (turns around): "See if this helps?"
Kid: "Ninety... Oh! You're a right angle!"
Me (turns back around): "Does this look like a right angle? I am an angle, but what kind?"
Kid: "I don't know. It says 90 degrees on your sign"
Me: "Look again. It doesn't just say 90 degrees"
Kid: "Yes it does. You're a right angle."
Me (to myself): "I'm not a right angle, but apparently I am the worst geometry teacher ever."

So I learned that about half my kids do not know what an acute angle is and/or do not know how to read greater than/less than signs. This is a problem. I'm okay with them not getting the pun, but seriously, you think that's a right angle? Whatever it is that I think I'm teaching is clearly not sticking. Scariest Halloween ever. I will be forever haunted by the terror of our nation's poor math education.